\newproblem{lay:6_7_1}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.7.1}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $\mathbf{u}=(u_1,u_2)$ and $\mathbf{v}=(v_1,v_2)$ be two vectors in $\mathbb{R}^2$. Let us define the inner product in $\mathbb{R}^2$ as
	\begin{center}
		$\mathbf{u}\cdot\mathbf{v}=4u_1v_1+5u_2v_2$
	\end{center}
	Let $\mathbf{x}=(1,1)$ and $\mathbf{y}=(5,-1)$.
	\begin{enumerate}[a.]
		\item Find $\|\mathbf{x}\|$, $\|\mathbf{y}\|$ and $|\mathbf{x}\cdot\mathbf{y}|^2$.
		\item Describe all vectors that are orthogonal to $\mathbf{y}$.
	\end{enumerate}
}{
   % Solution
	\begin{enumerate}[a.]
		\item To find the required quantities we use their definition in terms of the dot product
				  \begin{center}
						$\|\mathbf{x}\|=\sqrt{\mathbf{x}\cdot\mathbf{x}}=\sqrt{(1,1)\cdot(1,1)}=\sqrt{4\cdot 1\cdot 1+5\cdot 1\cdot 1}=\sqrt{9}$ \\
						$\|\mathbf{y}\|=\sqrt{\mathbf{y}\cdot\mathbf{y}}=\sqrt{(5,-1)\cdot(5,-1)}=\sqrt{4\cdot 5\cdot 5+5\cdot (-1)\cdot (-1)}=\sqrt{105}$ \\
						$\mathbf{x}\cdot\mathbf{y}=(1,1)\cdot(5,-1)=4\cdot 1\cdot 5+5\cdot 1\cdot (-1)=15$\\
						$|\mathbf{x}\cdot\mathbf{y}|^2=|15|^2=225$
					\end{center}
		\item Let $\mathbf{w}=(w_1,w_2)$ be an arbitrary vector in $\mathbb{R}^2$ orthogonal to $\mathbf{y}$. It must fulfill
		      \begin{center}
						$\mathbf{w}\cdot\mathbf{y}=0$\\
						$\mathbf{w}\cdot(5,-1)=0$\\
						$4w_1(5)+5w_2(-1)=0$\\
						$w_2=4w_1$\\
					\end{center}
					So, any vector $\mathbf{w}$ orthogonal to $\mathbf{y}$ according to the proposed inner product must be of the form $(w_1,4w_1)$.
	\end{enumerate}
}
\useproblem{lay:6_7_1}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

